A019519 -id:A019519 - cp's OEIS frontend

A078274 a(n) is the number from which if odd-positioned digits are deleted then one gets the concatenation of first n odd numbers, while if even-positioned digits are deleted then one gets reverse concatenation of first n odd numbers. Position of least significant digit is considered to be 1.

11, 1331, 153351, 17355371, 1937557391, 11315977951311, 113351719917151331, 1135517391111917351351, 11375175911311113917551371, 113951779115111331115917751391, 1231517991171115311351117917952311, 12335271911911173115511371119927152331

Offset: 1

Amarnath Murthy, Nov 25 2002

a(n) is formed by interleaving the digits of A019519(n) and A038395(n). - Sean A. Irvine, Jun 26 2025

			a(4) = 17355371: deleting alternate digits starting from the LSD gives 1357. Deleting the other digits gives 7531.

Cf. A078275, A078276, A078277, A078278.

Cf. A019519, A038395.

a(n) = {my(d, v=w=[1]); for(i=2, n, v=concat(v, d=digits(2*i-1)); w=concat(d, w)); fromdigits(vector(2*#v, i, if(i%2, v[1+i\2], w[i/2]))); } \\ Sean A. Irvine, Jun 26 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
Corrected and extended by Harvey P. Dale, Apr 11 2012
Original terms restored by Sean A. Irvine, Jun 26 2025

A138965 Least prime factor of concatenation of first n odd numbers.

Original entry on oeis.org

1, 13, 3, 23, 37, 3, 11617, 5, 3, 135791113151719, 29, 3, 5, 11, 3, 135791113151719212325272931, 17, 3, 7, 13, 3, 131, 5, 3, 11, 25471443030907588399109, 3, 5, 7, 3, 181, 41, 3, 135791113151719212325272931333537394143454749515355575961636567, 19, 3, 40351, 5, 3, 7, 11, 3, 5, 57041, 3, 351269, 11, 3, 135791113151719212325272931333537394143454749515355575961636567697173757779818385878991939597

Offset: 1

M. F. Hasler, Apr 14 2008

Sean A. Irvine, Table of n, a(n) for n = 1..300
F. Smarandache, Sequences of numbers involved in unsolved problems, arXiv:math/0604019

Cf. A019519, A046036, A048847; A109837, A104644.

t=1; for( n=2,99, print1( factor( eval( t=Str( t,2*n-1 )))[1,1], ", "))

A138965(n) = A020639(A019519(n)) (= 3 if n = 0 (mod 3)).

A349960 Values taken by the function A067095 in the order of their appearance.

Original entry on oeis.org

2, 1, 18, 181, 1817, 18175, 181757, 1817571, 18175719, 181757197, 1817571972, 18175719727, 181757197277, 1817571972772, 18175719727727, 181757197277277, 1817571972772779, 18175719727727795, 181757197277277957, 1817571972772779572, 18175719727727795720, 181757197277277957202

Offset: 1

Bernard Schott, Dec 07 2021

a(2) < a(1), but thereafter this function increases monotonically without limit (see Krusemeyer reference).
The record values > 2 of A067095(m) occur when m = 5, 50, 500, 5000, .... This happens precisely when the corresponding numerator A019520(m) goes from 2/4/6/8/10/12/....../999...98 to 2/4/6/8/10/12/....../999...98/1000...00, where here / means concatenation.
If a(n) is a k-digit number (k = A055642(a(n))), then 1.8 * 10^(k-1) < a(n) < 1.9 * 10^(k-1).
If we consider the sequence u(n) = a(n)/10^(k-1) where k = length(a(n)); we have u(n) is increasing with an upper bound 1.9; so, this sequence u(n) is convergent and, conjecture, this limit = 1.81757197277277957... found by Giorgos Kalogeropoulos; now, from this limit, it is possible to get the successive terms of this sequence here.

			Floor(A019520(5)/A019519(5)) = floor(246810/13579) = floor(18.175859...) = 18, hence, 18 is a term.

Mark I. Krusemeyer, George T. Gilbert and Loren C. Larson, A Mathematical Orchard, Problems and Solutions, MAA, 2012, Problem 87, pp. 159-161.

Chai Wah Wu, Table of n, a(n) for n = 1..1001

Cf. A019519, A019520, A055642, A067095.

terms=5; f[i_]:=FromDigits@Flatten[IntegerDigits/@i];
k[q_]:=f[Range[2,2q,2]]/f[Range[1,2q,2]];
DeleteDuplicates@Table[Floor[k@n],{n,10^(terms-2)/2}] (* Giorgos Kalogeropoulos, Dec 10 2021 *)

def A349960(n): return 3-n if n <= 2 else int("".join(str(d) for d in range(2,10**(n-2)+1,2)))//int("".join(str(d) for d in range(1,10**(n-2),2))) # Chai Wah Wu, Dec 10 2021
from itertools import count
def A349960(n): # a more efficient implementation
    if n <= 2:
        return 3-n
    a, b = '', ''
    for i in count(1,2):
        a += str(i)
        b += str(i+1)
        ai, bi = int(a), int(b)
        if len(a)+n-2 == len(b): return bi//ai
        m = 10**(n-2-len(b)+len(a))
        lb = bi*m//(ai+1)
        ub = (bi+1)*m//ai
        if lb == ub: return lb # Chai Wah Wu, Dec 10 2021

a(n) = floor(k((n + 6)/2)*10^(n - 1 - ceiling(log_10(k((n + 6)/2))))) for k(n) = A019520(n)/A019519(n) and n >= 2 (conjectured). - Giorgos Kalogeropoulos, Dec 10 2021

a(5)-a(7) from Michel Marcus, Dec 07 2021
a(8)-a(9) from Martin Ehrenstein, Dec 10 2021
a(10)-a(22) from Chai Wah Wu, Dec 10 2021

A078259 a(n) = denominator(N), where N = 0.135..(2n-1) is the concatenation of the first n odd numbers after decimal point.

Original entry on oeis.org

10, 100, 200, 10000, 100000, 10000000, 1000000000, 20000000000, 10000000000000, 1000000000000000, 100000000000000000, 10000000000000000000, 40000000000000000000, 100000000000000000000000, 10000000000000000000000000, 1000000000000000000000000000

Offset: 1

Amarnath Murthy, Nov 24 2002

Cf. A019519, A031312.

a:= n-> (t-> denom(t/10^length(t)))(parse(cat(2*i-1$i=1..n))):
seq(a(n), n=1..16);  # Alois P. Heinz, Jun 25 2025

Corrected and extended by Sascha Kurz, Jan 04 2003

a(4)-a(5) corrected and more terms from Sean A. Irvine, Jun 25 2025

cp's OEIS Frontend

A078274 a(n) is the number from which if odd-positioned digits are deleted then one gets the concatenation of first n odd numbers, while if even-positioned digits are deleted then one gets reverse concatenation of first n odd numbers. Position of least significant digit is considered to be 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A138965 Least prime factor of concatenation of first n odd numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A349960 Values taken by the function A067095 in the order of their appearance.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A078259 a(n) = denominator(N), where N = 0.135..(2n-1) is the concatenation of the first n odd numbers after decimal point.

Views

Author

Keywords

Crossrefs

Programs

Maple

Extensions